## TPTP Problem File: NUM793^4.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : NUM793^4 : TPTP v8.1.2. Released v7.1.0.
% Domain   : Number theory
% Problem  : Grundlagen problem satz87d
% Version  : [Bro17] axioms : Especial.
% English  :

% Refs     : [Bro17] Brown (2017), Email to G. Sutcliffe
% Source   : [Br017]
% Names    :

% Status   : Theorem
% Rating   : 0.85 v8.1.0, 0.82 v7.5.0, 0.86 v7.4.0, 0.89 v7.2.0, 0.88 v7.1.0
% Syntax   : Number of formulae    :  707 ( 208 unt; 199 typ; 192 def)
%            Number of atoms       : 2986 ( 222 equ;   0 cnn)
%            Maximal formula atoms :   16 (   5 avg)
%            Number of connectives : 6515 (   7   ~;   4   |;  14   &;6142   @)
%                                         (   3 <=>; 345  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   7 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :  580 ( 580   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  223 ( 221 usr;  35 con; 0-7 aty)
%            Number of variables   : 2022 (1863   ^ 151   !;   8   ?;2022   :)
% SPC      : TH0_THM_EQU_NAR

%------------------------------------------------------------------------------
include('Axioms/NUM007^0.ax').
include('Axioms/NUM007^1.ax').
%------------------------------------------------------------------------------
thf(typ_inf,type,
inf: \$i > \$i > \$o ).

thf(def_inf,definition,
( inf
= ( esti @ frac ) ) ).

thf(typ_rat,type,
rat: \$i ).

thf(def_rat,definition,
( rat
= ( ect @ frac @ n_eq ) ) ).

thf(typ_rt_is,type,
rt_is: \$i > \$i > \$o ).

thf(def_rt_is,definition,
( rt_is
= ( e_is @ rat ) ) ).

thf(typ_rt_nis,type,
rt_nis: \$i > \$i > \$o ).

thf(def_rt_nis,definition,
( rt_nis
= ( ^ [X0: \$i,X1: \$i] : ( d_not @ ( rt_is @ X0 @ X1 ) ) ) ) ).

thf(typ_rt_some,type,
rt_some: ( \$i > \$o ) > \$o ).

thf(def_rt_some,definition,
( rt_some
= ( l_some @ rat ) ) ).

thf(typ_rt_all,type,
rt_all: ( \$i > \$o ) > \$o ).

thf(def_rt_all,definition,
( rt_all
= ( all @ rat ) ) ).

thf(typ_rt_one,type,
rt_one: ( \$i > \$o ) > \$o ).

thf(def_rt_one,definition,
( rt_one
= ( one @ rat ) ) ).

thf(typ_rt_in,type,
rt_in: \$i > \$i > \$o ).

thf(def_rt_in,definition,
( rt_in
= ( esti @ rat ) ) ).

thf(typ_ratof,type,
ratof: \$i > \$i ).

thf(def_ratof,definition,
( ratof
= ( ectelt @ frac @ n_eq ) ) ).

thf(typ_class,type,
class: \$i > \$i ).

thf(def_class,definition,
( class
= ( ecect @ frac @ n_eq ) ) ).

thf(typ_fixf,type,
fixf: \$i > \$i > \$o ).

thf(def_fixf,definition,
( fixf
= ( fixfu2 @ frac @ n_eq ) ) ).

thf(typ_indrat,type,
indrat: \$i > \$i > \$i > \$i > \$i ).

thf(def_indrat,definition,
( indrat
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] : ( indeq2 @ frac @ n_eq @ X2 @ X3 @ X0 @ X1 ) ) ) ).

thf(satz78,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] : ( rt_is @ X0 @ X0 ) ) ).

thf(satz79,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_is @ X0 @ X1 )
=> ( rt_is @ X1 @ X0 ) ) ) ) ).

thf(satz80,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] : ( in @ X2 @ rat )
@ ^ [X2: \$i] :
( ( rt_is @ X0 @ X1 )
=> ( ( rt_is @ X1 @ X2 )
=> ( rt_is @ X0 @ X2 ) ) ) ) ) ) ).

thf(typ_rt_more,type,
rt_more: \$i > \$i > \$o ).

thf(def_rt_more,definition,
( rt_more
= ( ^ [X0: \$i,X1: \$i] :
( l_some @ frac
@ ^ [X2: \$i] :
( l_some @ frac
@ ^ [X3: \$i] : ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( moref @ X2 @ X3 ) ) ) ) ) ) ).

thf(typ_propm,type,
propm: \$i > \$i > \$i > \$i > \$o ).

thf(def_propm,definition,
( propm
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] : ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( moref @ X2 @ X3 ) ) ) ) ).

thf(typ_rt_less,type,
rt_less: \$i > \$i > \$o ).

thf(def_rt_less,definition,
( rt_less
= ( ^ [X0: \$i,X1: \$i] :
( l_some @ frac
@ ^ [X2: \$i] :
( l_some @ frac
@ ^ [X3: \$i] : ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( lessf @ X2 @ X3 ) ) ) ) ) ) ).

thf(typ_propl,type,
propl: \$i > \$i > \$i > \$i > \$o ).

thf(def_propl,definition,
( propl
= ( ^ [X0: \$i,X1: \$i,X2: \$i,X3: \$i] : ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( lessf @ X2 @ X3 ) ) ) ) ).

thf(satz81,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] : ( orec3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) ) ).

thf(satz81a,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] : ( or3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) ) ).

thf(satz81b,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] : ( ec3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) ) ).

thf(satz82,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_more @ X0 @ X1 )
=> ( rt_less @ X1 @ X0 ) ) ) ) ).

thf(satz83,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_less @ X0 @ X1 )
=> ( rt_more @ X1 @ X0 ) ) ) ) ).

thf(typ_rt_moreis,type,
rt_moreis: \$i > \$i > \$o ).

thf(def_rt_moreis,definition,
( rt_moreis
= ( ^ [X0: \$i,X1: \$i] : ( l_or @ ( rt_more @ X0 @ X1 ) @ ( rt_is @ X0 @ X1 ) ) ) ) ).

thf(typ_rt_lessis,type,
rt_lessis: \$i > \$i > \$o ).

thf(def_rt_lessis,definition,
( rt_lessis
= ( ^ [X0: \$i,X1: \$i] : ( l_or @ ( rt_less @ X0 @ X1 ) @ ( rt_is @ X0 @ X1 ) ) ) ) ).

thf(satz81c,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_moreis @ X0 @ X1 )
=> ( d_not @ ( rt_less @ X0 @ X1 ) ) ) ) ) ).

thf(satz81d,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_lessis @ X0 @ X1 )
=> ( d_not @ ( rt_more @ X0 @ X1 ) ) ) ) ) ).

thf(satz81e,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( d_not @ ( rt_more @ X0 @ X1 ) )
=> ( rt_lessis @ X0 @ X1 ) ) ) ) ).

thf(satz81f,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( d_not @ ( rt_less @ X0 @ X1 ) )
=> ( rt_moreis @ X0 @ X1 ) ) ) ) ).

thf(satz81g,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_more @ X0 @ X1 )
=> ( d_not @ ( rt_lessis @ X0 @ X1 ) ) ) ) ) ).

thf(satz81h,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_less @ X0 @ X1 )
=> ( d_not @ ( rt_moreis @ X0 @ X1 ) ) ) ) ) ).

thf(satz81j,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( d_not @ ( rt_moreis @ X0 @ X1 ) )
=> ( rt_less @ X0 @ X1 ) ) ) ) ).

thf(satz81k,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( d_not @ ( rt_lessis @ X0 @ X1 ) )
=> ( rt_more @ X0 @ X1 ) ) ) ) ).

thf(satz84,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_moreis @ X0 @ X1 )
=> ( rt_lessis @ X1 @ X0 ) ) ) ) ).

thf(satz85,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( ( rt_lessis @ X0 @ X1 )
=> ( rt_moreis @ X1 @ X0 ) ) ) ) ).

thf(satz86,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] : ( in @ X2 @ rat )
@ ^ [X2: \$i] :
( ( rt_less @ X0 @ X1 )
=> ( ( rt_less @ X1 @ X2 )
=> ( rt_less @ X0 @ X2 ) ) ) ) ) ) ).

thf(satz87a,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] : ( in @ X2 @ rat )
@ ^ [X2: \$i] :
( ( rt_lessis @ X0 @ X1 )
=> ( ( rt_less @ X1 @ X2 )
=> ( rt_less @ X0 @ X2 ) ) ) ) ) ) ).

thf(satz87b,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] : ( in @ X2 @ rat )
@ ^ [X2: \$i] :
( ( rt_less @ X0 @ X1 )
=> ( ( rt_lessis @ X1 @ X2 )
=> ( rt_less @ X0 @ X2 ) ) ) ) ) ) ).

thf(satz87c,axiom,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] : ( in @ X2 @ rat )
@ ^ [X2: \$i] :
( ( rt_moreis @ X0 @ X1 )
=> ( ( rt_more @ X1 @ X2 )
=> ( rt_more @ X0 @ X2 ) ) ) ) ) ) ).

thf(satz87d,conjecture,
( all_of
@ ^ [X0: \$i] : ( in @ X0 @ rat )
@ ^ [X0: \$i] :
( all_of
@ ^ [X1: \$i] : ( in @ X1 @ rat )
@ ^ [X1: \$i] :
( all_of
@ ^ [X2: \$i] : ( in @ X2 @ rat )
@ ^ [X2: \$i] :
( ( rt_more @ X0 @ X1 )
=> ( ( rt_moreis @ X1 @ X2 )
=> ( rt_more @ X0 @ X2 ) ) ) ) ) ) ).

%------------------------------------------------------------------------------
```